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NCERT Class 10 Mathematics Exercise 10.1 Solutions: Circles
AI-written, checked on 23 Oct 2026
Solutions written by Shishya, checked — not the board's official answers
Step-by-step working for every question of Exercise 10.1 in NCERT's Class 10 Mathematics book, chapter “Circles”, written by Shishya. Each question is in our own short words; open the NCERT chapter for the exact wording and figures.
How these were checked: the solutions were drafted with AI and then read and corrected on 23 Oct 2026. Every numerical final answer is recomputed from the question's own numbers by a calculator check that runs before each release; proofs and explanations were read, not machine-checked. The NCERT textbook is the authority — if something here disagrees with it, follow the book.
Tap “why?” under a step to see the textbook rule it uses — from Shishya's fixed list of NCERT Class 9-10 rules, not written by AI for this page. A step without one has no rule from the list yet.
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Ex 10.1 Q1
How many tangents can a circle have?
- A tangent to a circle is a line that touches the circle at exactly one point.
- Since a circle is a closed curve, there are infinitely many points on its circumference.
- At each of these infinitely many points, exactly one tangent can be drawn.
- Therefore, a circle can have infinitely many tangents.
Answer: A circle can have infinitely many tangents.
Ex 10.1 Q2
Fill the blanks: (i) a tangent meets a circle in … point(s) (ii) a line meeting a circle in two points is a … (iii) a circle can have at most … parallel tangents (iv) the common point of a tangent and the circle is the …
- (i) A tangent touches the circle at exactly one location, so it meets the circle in one point.
- (ii) A line that intersects the circle at two distinct points is called a secant.
- (iii) A circle can have two parallel tangents, one on each side of the diameter.
- (iv) The specific point where the tangent line touches the circle is known as the point of contact.
Answer: (i) one; (ii) secant; (iii) two; (iv) point of contact
Ex 10.1 Q3
A tangent PQ at P of a circle of radius 5 cm meets a line through the centre O at Q, with OQ = 12 cm. Find PQ: (A) 12 cm (B) 13 cm (C) 8.5 cm (D) √119 cm.
- Since PQ is a tangent to the circle at point P, the radius OP is perpendicular to the tangent PQ.
why?
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
- Therefore, angle OPQ is 90 degrees, making triangle OPQ a right-angled triangle.
- Given that the radius OP is 5 cm and the distance OQ is 12 cm.
- Apply the Pythagoras theorem to triangle OPQ: OP² + PQ² = OQ².
why?
In a right-angled triangle, (hypotenuse)² = (one side)² + (other side)² (Baudhāyana–Pythagoras theorem). The hypotenuse is the side opposite the right angle and is the longest side.
- Substitute the known values into the equation: 5² + PQ² = 12².
why?
Putting the known values in place of the letters turns the formula or equation into plain arithmetic — the letters stood for exactly these values.
- Calculate the squares: 25 + PQ² = 144.
- Rearrange the equation to solve for PQ²: PQ² = 144 − 25.
- Compute the difference: PQ² = 119.
- Take the square root of both sides to find the length of PQ: PQ = √119 cm.
why?
If two non-negative quantities are equal, their square roots are equal. A square has two roots, + and −; a length or a count cannot be negative, so the negative root is dropped there.
- Compare this result with the given options to identify the correct choice.
Answer: (D) √119 cm
- Since PQ is a tangent to the circle at point P, the radius OP is perpendicular to the tangent PQ.
Ex 10.1 Q4
Draw a circle and two lines parallel to a given line so that one is a tangent and the other a secant.
- Draw a circle with centre O and radius 5 cm.
- Draw a line l outside the circle such that its distance from O is greater than 5 cm.
- From O, draw a perpendicular to line l meeting it at point P.
- Draw a line m parallel to l passing through P; this line is at a distance equal to the radius from O.
- Since the distance from O to m equals the radius, m touches the circle at exactly one point, making it a tangent.
why?
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
- Draw a line n parallel to l on the other side of O such that its distance from O is less than 5 cm.
- Since the distance from O to n is less than the radius, line n cuts the circle at two distinct points, making it a secant.
- Thus, m is a tangent and n is a secant, both parallel to the given line l.
Answer: A circle is drawn with a tangent line and a secant line, both parallel to a given line.
Other exercises in this chapter
- Exercise 10.2 — 13 questions
← Circles: notes, practice and all exercises
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